Applied Econometrics

Stationary and Non-stationary Time series

Stationary

During regression, we assume that the time series is stationary. A time series is stationary if its mean and variance are constant overtime A time series is an example of stochastic process- sequence of random variables ordered in time.

(Log of exchange rate)

LnEx is generally drifting upwards, though with great variation, suggesting that neither mean nor variance of this time series is constant, thus LnEX is non-stationary or unit root.

Example: Inflation rates

Henry and shields found that standard unit root tests such as dickey fuller and Elliot et al, and the KPSS tests of the null hypothesis of stationary may provide misleading evidence as to the degree of persistence of shocks to inflation.

If a time series is unit root, we can study its behavior only for the period under consideration. As a result it is not possible to generate for future time periods.

Tests of stationary: -

Graphical analysis (visually) Autocorrelation function ( correlation against itself) Dickey fuller test Augmented Fuller test Phillips Perron test

Autocorrelation function :- ACF at lag k is defined as:

pk = Yk/Y0 = covariance at lag k/variance

Akaike/Schwarz information criterion to determine the lag length. Rule of thumb is to find ACF up to ¼ to1/3 the length of the time series Test the stat. significance of each AC coefficient in the correlogram by calculating its standard

error Alternatively, find out if the sum of autpcorrelation coefficient is stat. significant. Using the Q

statistic where n is the sample size, and the m is the number of lags

Q= ∑i= k=1

k=m

pk2

E.g.

LAG AC PAC Q Prob 1 0.9985 1.0001 2350.9 0.0000 2 0.9970 -0.0021 4695.7 0.0000 3 0.9954 0.0009 7034.2 0.0000 4 0.9939 0.0169 9366.6 0.0000 5 0.9924 -0.0516 11693 0.0000

Dickey Fuller (DF) test

The unit root test for the variable is ∆Y t=B1+B2t+B3Y t−1+ut Regress the differences of the log of exchange rate on the trend variable and the on-period

lagged balue of the exchange rate. The null hypothesis is that B3, the coefficient of Y t−1, is zero. In the unit root context, the null hypothesis is Y contains a unit root The alternative hypothesis is Y is stationary. Use the DF test, whose critical values are calculated by simulations and modern stat packages,

such as EVIEWS and STATA. DF test requires a comparison of the Z(t) stat to the critical values, it must be lower the DF

critical values. If absolute test statistic of b3 (Z(t)) is less than absolute DF critical value, indicates nonrejection

of the unit root null. Absolute test stat on b3 (Z(t)) is 0.17, which is less than absolute DF critical value (1%) 3.43.

Hence, we can accept the null hypothesis that LEX does contain a unit root. DF test can be performed in three different forms:

o Random Walk: ∆Y t=B3Y t−1+uto Random Walk with a drift : ∆Y t=B1+B3Y t−1+uto Random walk with drift and a deterministic trend : ∆Y t=B1+B2t+B3Y t−1+ut

The simple DF test is only valid if the series is an AR(1) process. If the series is correlated at higher order lags, the assumption disturbances ut is violated.

ADF (augmented dickey-fuller) tests constructs a parametric correction for higher-order correlation by assuming that the series follows an AR (p) process and adding p lagged differences terms of the dependent variable (Y) to the right hand side of the test regression.

If the error term ut is correlated, use ADF. Add the lagged values of the dependent variables aka

∆Y t=B1+B2t+B3Y t−1+∑i=0

m

ai∆Y t−i+et

Null and alternative hypotheses are same as DF test. Phillips Perron test an alternative method of controlling for serial correlation when testing for a

unit root.

The PP method estimates the non-augmented DF test equation, and modifies the t-ratio of the a coefficient so that serial correlation does not affect the asymptotic distribution of the test statistic.

The PP test is based on the statistic:

~t a=t a( γ 0f 0 )1/2

−T ( f 0−γ 0 ) ( se ( a ) )

2 f 01/2 s Where the a is the estimate and the ta is the t-ratio of a, se(a) is coefficient standard error, and

the s is the standard error of the test regression, In addition, γ0 is consistent estimate of the error variance, f0 is an estimator of the residual spectrum at frequency zero.

There are two choices you will have make when performing the PP test: First, you must choose whether to include a constant, a constant a linear line trend, or neither in

the test regression. Second, you will have to choose a method for estimating f0. Stata and Eviews has estimators to

estimate this. PP test with trend. Null and alternative hypotheses are same as ADF test. PP test shows Y is a

unit root variable. If there is a general trend you can make it stationary by removing the trend from it. Yt = A1+A2t+et, where t is trend variable and e is error term with the usual properties. After running the regression, e=Y t−A1−A2t The estimated error term in the above equation now represents the detrended Y time series i.e.

Y without the trend. This procedure is valid if the original Y has a deterministic trend. If the time series becomes

stationary after detrending then it is called trend stationary. Taking the first difference Y is a better way to make it stationary.